Solve the following equations for \( x \). \( \begin{array}{ll}17 .) 5^{2 x}=5^{2} & 18 \\ \text { 19. }(2.5)^{2 x+1}=(2.5)^{5} & 20 \\ \text { 21. } 10^{1-x}=100 & 22 \\ \text { 23. } 3(2.7)^{5 x}=8.1 & 24 \\ \text { 25. }\left(2^{x+1} \cdot 2^{-3}\right)^{2}=2 & 26\end{array} \)

Solve the equation \( 5^{2x}=5^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{2x}=5^{2}\) - step1: Set the exponents equal: \(2x=2\) - step2: Divide both sides: \(\frac{2x}{2}=\frac{2}{2}\) - step3: Divide the numbers: \(x=1\) Solve the equation \( (2.5)^{2x+1}=(2.5)^{5} \). Solve the equation by following steps: - step0: Solve for \(x\): \(2.5^{2x+1}=2.5^{5}\) - step1: Set the exponents equal: \(2x+1=5\) - step2: Move the constant to the right side: \(2x=5-1\) - step3: Subtract the numbers: \(2x=4\) - step4: Divide both sides: \(\frac{2x}{2}=\frac{4}{2}\) - step5: Divide the numbers: \(x=2\) Solve the equation \( 10^{1-x}=100 \). Solve the equation by following steps: - step0: Solve for \(x\): \(10^{1-x}=100\) - step1: Rewrite in exponential form: \(10^{1-x}=10^{2}\) - step2: Set the exponents equal: \(1-x=2\) - step3: Move the constant to the right side: \(-x=2-1\) - step4: Subtract the numbers: \(-x=1\) - step5: Change the signs: \(x=-1\) Solve the equation \( 3(2.7)^{5x}=8.1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3\times 2.7^{5x}=8.1\) - step1: Convert the expressions: \(3\left(\frac{27}{10}\right)^{5x}=8.1\) - step2: Multiply by the reciprocal: \(3\left(\frac{27}{10}\right)^{5x}\times \frac{1}{3}=8.1\times \frac{1}{3}\) - step3: Multiply: \(\left(\frac{27}{10}\right)^{5x}=\frac{27}{10}\) - step4: Set the exponents equal: \(5x=1\) - step5: Divide both sides: \(\frac{5x}{5}=\frac{1}{5}\) - step6: Divide the numbers: \(x=\frac{1}{5}\) Solve the equation \( \left(2^{x+1} \cdot 2^{-3}\right)^{2}=2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left(2^{x+1}\times 2^{-3}\right)^{2}=2\) - step1: Simplify: \(2^{2\left(x-2\right)}=2\) - step2: Set the exponents equal: \(2\left(x-2\right)=1\) - step3: Divide both sides: \(\frac{2\left(x-2\right)}{2}=\frac{1}{2}\) - step4: Divide the numbers: \(x-2=\frac{1}{2}\) - step5: Move the constant to the right side: \(x=\frac{1}{2}+2\) - step6: Add the numbers: \(x=\frac{5}{2}\) The solutions to the given equations are: 1. \(5^{2x}=5^{2}\) has the solution \(x=1\). 2. \((2.5)^{2x+1}=(2.5)^{5}\) has the solution \(x=2\). 3. \(10^{1-x}=100\) has the solution \(x=-1\). 4. \(3(2.7)^{5x}=8.1\) has the solution \(x=\frac{1}{5}\) or \(x=0.2\). 5. \(\left(2^{x+1} \cdot 2^{-3}\right)^{2}=2\) has the solution \(x=\frac{5}{2}\) or \(x=2.5\).